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## G = C3×C52⋊C4order 300 = 22·3·52

### Direct product of C3 and C52⋊C4

Aliases: C3×C52⋊C4, C154F5, C526C12, (C5×C15)⋊8C4, C52(C3×F5), C5⋊D5.3C6, (C3×C5⋊D5).4C2, SmallGroup(300,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C3×C52⋊C4
 Chief series C1 — C5 — C52 — C5⋊D5 — C3×C5⋊D5 — C3×C52⋊C4
 Lower central C52 — C3×C52⋊C4
 Upper central C1 — C3

Generators and relations for C3×C52⋊C4
G = < a,b,c,d | a3=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >

25C2
2C5
2C5
25C4
25C6
5D5
5D5
10D5
10D5
2C15
2C15
25C12
5F5
5F5
10C3×D5
10C3×D5

Character table of C3×C52⋊C4

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 5E 5F 6A 6B 12A 12B 12C 12D 15A 15B 15C 15D 15E 15F 15G 15H 15I 15J 15K 15L size 1 25 1 1 25 25 4 4 4 4 4 4 25 25 25 25 25 25 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 ζ3 ζ32 -1 -1 1 1 1 1 1 1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ4 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 1 ζ32 ζ3 -1 -1 1 1 1 1 1 1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 -1 1 1 -i i 1 1 1 1 1 1 -1 -1 -i i -i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 -1 1 1 i -i 1 1 1 1 1 1 -1 -1 i -i i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 1 -1 ζ3 ζ32 -i i 1 1 1 1 1 1 ζ65 ζ6 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 12 ρ10 1 -1 ζ32 ζ3 i -i 1 1 1 1 1 1 ζ6 ζ65 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 12 ρ11 1 -1 ζ3 ζ32 i -i 1 1 1 1 1 1 ζ65 ζ6 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 12 ρ12 1 -1 ζ32 ζ3 -i i 1 1 1 1 1 1 ζ6 ζ65 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 12 ρ13 4 0 4 4 0 0 -1 -1 4 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 4 -1 4 orthogonal lifted from F5 ρ14 4 0 4 4 0 0 -1 4 -1 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 4 -1 4 -1 orthogonal lifted from F5 ρ15 4 0 4 4 0 0 3+√5/2 -1 -1 3-√5/2 -1-√5 -1+√5 0 0 0 0 0 0 3+√5/2 3-√5/2 -1-√5 -1+√5 3+√5/2 3-√5/2 -1-√5 -1+√5 -1 -1 -1 -1 orthogonal lifted from C52⋊C4 ρ16 4 0 4 4 0 0 -1-√5 -1 -1 -1+√5 3-√5/2 3+√5/2 0 0 0 0 0 0 -1-√5 -1+√5 3-√5/2 3+√5/2 -1-√5 -1+√5 3-√5/2 3+√5/2 -1 -1 -1 -1 orthogonal lifted from C52⋊C4 ρ17 4 0 4 4 0 0 -1+√5 -1 -1 -1-√5 3+√5/2 3-√5/2 0 0 0 0 0 0 -1+√5 -1-√5 3+√5/2 3-√5/2 -1+√5 -1-√5 3+√5/2 3-√5/2 -1 -1 -1 -1 orthogonal lifted from C52⋊C4 ρ18 4 0 4 4 0 0 3-√5/2 -1 -1 3+√5/2 -1+√5 -1-√5 0 0 0 0 0 0 3-√5/2 3+√5/2 -1+√5 -1-√5 3-√5/2 3+√5/2 -1+√5 -1-√5 -1 -1 -1 -1 orthogonal lifted from C52⋊C4 ρ19 4 0 -2+2√-3 -2-2√-3 0 0 -1 4 -1 -1 -1 -1 0 0 0 0 0 0 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 -2+2√-3 ζ65 -2-2√-3 ζ6 complex lifted from C3×F5 ρ20 4 0 -2+2√-3 -2-2√-3 0 0 -1 -1 4 -1 -1 -1 0 0 0 0 0 0 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 ζ65 -2+2√-3 ζ6 -2-2√-3 complex lifted from C3×F5 ρ21 4 0 -2-2√-3 -2+2√-3 0 0 -1 -1 4 -1 -1 -1 0 0 0 0 0 0 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 ζ6 -2-2√-3 ζ65 -2+2√-3 complex lifted from C3×F5 ρ22 4 0 -2-2√-3 -2+2√-3 0 0 -1 4 -1 -1 -1 -1 0 0 0 0 0 0 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 -2-2√-3 ζ6 -2+2√-3 ζ65 complex lifted from C3×F5 ρ23 4 0 -2-2√-3 -2+2√-3 0 0 3-√5/2 -1 -1 3+√5/2 -1+√5 -1-√5 0 0 0 0 0 0 ζ3ζ53+ζ3ζ52+2ζ3 ζ3ζ54+ζ3ζ5+2ζ3 2ζ3ζ54+2ζ3ζ5 2ζ3ζ53+2ζ3ζ52 ζ32ζ53+ζ32ζ52+2ζ32 ζ32ζ54+ζ32ζ5+2ζ32 2ζ32ζ54+2ζ32ζ5 2ζ32ζ53+2ζ32ζ52 ζ6 ζ6 ζ65 ζ65 complex faithful ρ24 4 0 -2+2√-3 -2-2√-3 0 0 -1+√5 -1 -1 -1-√5 3+√5/2 3-√5/2 0 0 0 0 0 0 2ζ32ζ54+2ζ32ζ5 2ζ32ζ53+2ζ32ζ52 ζ32ζ54+ζ32ζ5+2ζ32 ζ32ζ53+ζ32ζ52+2ζ32 2ζ3ζ54+2ζ3ζ5 2ζ3ζ53+2ζ3ζ52 ζ3ζ54+ζ3ζ5+2ζ3 ζ3ζ53+ζ3ζ52+2ζ3 ζ65 ζ65 ζ6 ζ6 complex faithful ρ25 4 0 -2-2√-3 -2+2√-3 0 0 -1+√5 -1 -1 -1-√5 3+√5/2 3-√5/2 0 0 0 0 0 0 2ζ3ζ54+2ζ3ζ5 2ζ3ζ53+2ζ3ζ52 ζ3ζ54+ζ3ζ5+2ζ3 ζ3ζ53+ζ3ζ52+2ζ3 2ζ32ζ54+2ζ32ζ5 2ζ32ζ53+2ζ32ζ52 ζ32ζ54+ζ32ζ5+2ζ32 ζ32ζ53+ζ32ζ52+2ζ32 ζ6 ζ6 ζ65 ζ65 complex faithful ρ26 4 0 -2+2√-3 -2-2√-3 0 0 3-√5/2 -1 -1 3+√5/2 -1+√5 -1-√5 0 0 0 0 0 0 ζ32ζ53+ζ32ζ52+2ζ32 ζ32ζ54+ζ32ζ5+2ζ32 2ζ32ζ54+2ζ32ζ5 2ζ32ζ53+2ζ32ζ52 ζ3ζ53+ζ3ζ52+2ζ3 ζ3ζ54+ζ3ζ5+2ζ3 2ζ3ζ54+2ζ3ζ5 2ζ3ζ53+2ζ3ζ52 ζ65 ζ65 ζ6 ζ6 complex faithful ρ27 4 0 -2-2√-3 -2+2√-3 0 0 -1-√5 -1 -1 -1+√5 3-√5/2 3+√5/2 0 0 0 0 0 0 2ζ3ζ53+2ζ3ζ52 2ζ3ζ54+2ζ3ζ5 ζ3ζ53+ζ3ζ52+2ζ3 ζ3ζ54+ζ3ζ5+2ζ3 2ζ32ζ53+2ζ32ζ52 2ζ32ζ54+2ζ32ζ5 ζ32ζ53+ζ32ζ52+2ζ32 ζ32ζ54+ζ32ζ5+2ζ32 ζ6 ζ6 ζ65 ζ65 complex faithful ρ28 4 0 -2+2√-3 -2-2√-3 0 0 3+√5/2 -1 -1 3-√5/2 -1-√5 -1+√5 0 0 0 0 0 0 ζ32ζ54+ζ32ζ5+2ζ32 ζ32ζ53+ζ32ζ52+2ζ32 2ζ32ζ53+2ζ32ζ52 2ζ32ζ54+2ζ32ζ5 ζ3ζ54+ζ3ζ5+2ζ3 ζ3ζ53+ζ3ζ52+2ζ3 2ζ3ζ53+2ζ3ζ52 2ζ3ζ54+2ζ3ζ5 ζ65 ζ65 ζ6 ζ6 complex faithful ρ29 4 0 -2+2√-3 -2-2√-3 0 0 -1-√5 -1 -1 -1+√5 3-√5/2 3+√5/2 0 0 0 0 0 0 2ζ32ζ53+2ζ32ζ52 2ζ32ζ54+2ζ32ζ5 ζ32ζ53+ζ32ζ52+2ζ32 ζ32ζ54+ζ32ζ5+2ζ32 2ζ3ζ53+2ζ3ζ52 2ζ3ζ54+2ζ3ζ5 ζ3ζ53+ζ3ζ52+2ζ3 ζ3ζ54+ζ3ζ5+2ζ3 ζ65 ζ65 ζ6 ζ6 complex faithful ρ30 4 0 -2-2√-3 -2+2√-3 0 0 3+√5/2 -1 -1 3-√5/2 -1-√5 -1+√5 0 0 0 0 0 0 ζ3ζ54+ζ3ζ5+2ζ3 ζ3ζ53+ζ3ζ52+2ζ3 2ζ3ζ53+2ζ3ζ52 2ζ3ζ54+2ζ3ζ5 ζ32ζ54+ζ32ζ5+2ζ32 ζ32ζ53+ζ32ζ52+2ζ32 2ζ32ζ53+2ζ32ζ52 2ζ32ζ54+2ζ32ζ5 ζ6 ζ6 ζ65 ζ65 complex faithful

Permutation representations of C3×C52⋊C4
On 30 points - transitive group 30T73
Generators in S30
(1 13 8)(2 14 9)(3 15 10)(4 11 6)(5 12 7)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 20 19 18 17)(21 25 24 23 22)(26 30 29 28 27)
(1 16)(2 19 5 18)(3 17 4 20)(6 25 10 22)(7 23 9 24)(8 21)(11 30 15 27)(12 28 14 29)(13 26)

G:=sub<Sym(30)| (1,13,8)(2,14,9)(3,15,10)(4,11,6)(5,12,7)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27), (1,16)(2,19,5,18)(3,17,4,20)(6,25,10,22)(7,23,9,24)(8,21)(11,30,15,27)(12,28,14,29)(13,26)>;

G:=Group( (1,13,8)(2,14,9)(3,15,10)(4,11,6)(5,12,7)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27), (1,16)(2,19,5,18)(3,17,4,20)(6,25,10,22)(7,23,9,24)(8,21)(11,30,15,27)(12,28,14,29)(13,26) );

G=PermutationGroup([[(1,13,8),(2,14,9),(3,15,10),(4,11,6),(5,12,7),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,20,19,18,17),(21,25,24,23,22),(26,30,29,28,27)], [(1,16),(2,19,5,18),(3,17,4,20),(6,25,10,22),(7,23,9,24),(8,21),(11,30,15,27),(12,28,14,29),(13,26)]])

G:=TransitiveGroup(30,73);

Matrix representation of C3×C52⋊C4 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 60 1 0 0 16 44 0 0 59 44 18 18 17 1 43 60
,
 0 18 0 0 44 17 0 0 44 17 0 1 1 18 60 43
,
 0 0 60 1 18 1 59 43 0 0 60 0 17 1 60 0
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[60,16,59,17,1,44,44,1,0,0,18,43,0,0,18,60],[0,44,44,1,18,17,17,18,0,0,0,60,0,0,1,43],[0,18,0,17,0,1,0,1,60,59,60,60,1,43,0,0] >;

C3×C52⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_5^2\rtimes C_4
% in TeX

G:=Group("C3xC5^2:C4");
// GroupNames label

G:=SmallGroup(300,31);
// by ID

G=gap.SmallGroup(300,31);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-5,30,723,173,3004,1014]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;
// generators/relations

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